Journal of Machine Learning Research 3 (2003) 993-1022 Submitted 2/02 Published 1/03 Latent Dirichlet Allocation David M. Blei BLEI@CS.BERKELEY.EDU Computer Science Division University of California Berkeley, CA 94720, USA Andrew Y. Ng ANG@CS.STANFORD.EDU Computer Science Department Stanford University Stanford, CA 94305, USA Michael I. Jordan JORDAN@CS.BERKELEY.EDU Computer Science Division and Department of Statistics University of California Berkeley, CA 94720, USA Editor: John Lafferty Abstract We describe latent Dirichlet allocation (LDA), a generative probabilistic model for collections of discrete data such as text corpora. LDA is a three-level hierarchical Bayesian model, in which each item of a collection is modeled as a finite mixture over an underlying set of topics. Each topic is, in turn, modeled as an infinite mixture over an underlying set of topic probabilities. In the context of text modeling, the topic probabilities provide an explicit representation of a document. We present efficient approximate inference techniques based on variational methods and an EM algorithm for empirical Bayes parameter estimation. We report results in document modeling, text classification, and collaborative filtering, comparing to a mixture of unigrams model and the probabilistic LSI model. 1. Introduction In this paper we consider the problem of modeling text corpora and other collections of discrete data. The goal is to find short descriptions of the members of a collection that enable efficient processing of large collections while preserving the essential statistical relationships that are useful for basic tasks such as classification, novelty detection, summarization, and similarity and relevance judgments. Significant progress has been made on this problem by researchers in the field of informa- tion retrieval (IR) (Baeza-Yates and Ribeiro-Neto, 1999). The basic methodology proposed by IR researchers for text corpora���a methodology successfully deployed in modern Internet search engines���reduces each document in the corpus to a vector of real numbers, each of which repre- sents ratios of counts. In the popular tf-idf scheme (Salton and McGill, 1983), a basic vocabulary of ���words��� or ���terms��� is chosen, and, for each document in the corpus, a count is formed of the number of occurrences of each word. After suitable normalization, this term frequency count is compared to an inverse document frequency count, which measures the number of occurrences of a c 2003 David M. Blei, Andrew Y. Ng and Michael I. Jordan.

BLEI, NG, AND JORDAN word in the entire corpus (generally on a log scale, and again suitably normalized). The end result is a term-by-document matrix X whose columns contain the tf-idf values for each of the documents in the corpus. Thus the tf-idf scheme reduces documents of arbitrary length to fixed-length lists of numbers. While the tf-idf reduction has some appealing features���notably in its basic identification of sets of words that are discriminative for documents in the collection���the approach also provides a rela- tively small amount of reduction in description length and reveals little in the way of inter- or intra- document statistical structure. To address these shortcomings, IR researchers have proposed several other dimensionality reduction techniques, most notably latent semantic indexing (LSI) (Deerwester et al., 1990). LSI uses a singular value decomposition of the X matrix to identify a linear subspace in the space of tf-idf features that captures most of the variance in the collection. This approach can achieve significant compression in large collections. Furthermore, Deerwester et al. argue that the derived features of LSI, which are linear combinations of the original tf-idf features, can capture some aspects of basic linguistic notions such as synonymy and polysemy. To substantiate the claims regarding LSI, and to study its relative strengths and weaknesses, it is useful to develop a generative probabilistic model of text corpora and to study the ability of LSI to recover aspects of the generative model from data (Papadimitriou et al., 1998). Given a generative model of text, however, it is not clear why one should adopt the LSI methodology���one can attempt to proceed more directly, fitting the model to data using maximum likelihood or Bayesian methods. A significant step forward in this regard was made by Hofmann (1999), who presented the probabilistic LSI (pLSI) model, also known as the aspect model, as an alternative to LSI. The pLSI approach, which we describe in detail in Section 4.3, models each word in a document as a sample from a mixture model, where the mixture components are multinomial random variables that can be viewed as representations of ���topics.��� Thus each word is generated from a single topic, and different words in a document may be generated from different topics. Each document is represented as a list of mixing proportions for these mixture components and thereby reduced to a probability distribution on a fixed set of topics. This distribution is the ���reduced description��� associated with the document. While Hofmann���s work is a useful step toward probabilistic modeling of text, it is incomplete in that it provides no probabilistic model at the level of documents. In pLSI, each document is represented as a list of numbers (the mixing proportions for topics), and there is no generative probabilistic model for these numbers. This leads to several problems: (1) the number of parame- ters in the model grows linearly with the size of the corpus, which leads to serious problems with overfitting, and (2) it is not clear how to assign probability to a document outside of the training set. To see how to proceed beyond pLSI, let us consider the fundamental probabilistic assumptions underlying the class of dimensionality reduction methods that includes LSI and pLSI. All of these methods are based on the ���bag-of-words��� assumption���that the order of words in a document can be neglected. In the language of probability theory, this is an assumption of exchangeability for the words in a document (Aldous, 1985). Moreover, although less often stated formally, these methods also assume that documents are exchangeable the specific ordering of the documents in a corpus can also be neglected. A classic representation theorem due to de Finetti (1990) establishes that any collection of ex- changeable random variables has a representation as a mixture distribution���in general an infinite mixture. Thus, if we wish to consider exchangeable representations for documents and words, we need to consider mixture models that capture the exchangeability of both words and documents. 994

LATENT DIRICHLET ALLOCATION This line of thinking leads to the latent Dirichlet allocation (LDA) model that we present in the current paper. It is important to emphasize that an assumption of exchangeability is not equivalent to an as- sumption that the random variables are independent and identically distributed. Rather, exchange- ability essentially can be interpreted as meaning ���conditionally independent and identically dis- tributed,��� where the conditioning is with respect to an underlying latent parameter of a probability distribution. Conditionally, the joint distribution of the random variables is simple and factored while marginally over the latent parameter, the joint distribution can be quite complex. Thus, while an assumption of exchangeability is clearly a major simplifying assumption in the domain of text modeling, and its principal justification is that it leads to methods that are computationally efficient, the exchangeability assumptions do not necessarily lead to methods that are restricted to simple frequency counts or linear operations. We aim to demonstrate in the current paper that, by taking the de Finetti theorem seriously, we can capture significant intra-document statistical structure via the mixing distribution. It is also worth noting that there are a large number of generalizations of the basic notion of exchangeability, including various forms of partial exchangeability, and that representation theo- rems are available for these cases as well (Diaconis, 1988). Thus, while the work that we discuss in the current paper focuses on simple ���bag-of-words��� models, which lead to mixture distributions for single words (unigrams), our methods are also applicable to richer models that involve mixtures for larger structural units such as n-grams or paragraphs. The paper is organized as follows. In Section 2 we introduce basic notation and terminology. The LDA model is presented in Section 3 and is compared to related latent variable models in Section 4. We discuss inference and parameter estimation for LDA in Section 5. An illustrative example of fitting LDA to data is provided in Section 6. Empirical results in text modeling, text classification and collaborative filtering are presented in Section 7. Finally, Section 8 presents our conclusions. 2. Notation and terminology We use the language of text collections throughout the paper, referring to entities such as ���words,��� ���documents,��� and ���corpora.��� This is useful in that it helps to guide intuition, particularly when we introduce latent variables which aim to capture abstract notions such as topics. It is important to note, however, that the LDA model is not necessarily tied to text, and has applications to other problems involving collections of data, including data from domains such as collaborative filtering, content-based image retrieval and bioinformatics. Indeed, in Section 7.3, we present experimental results in the collaborative filtering domain. Formally, we define the following terms: ��� A word is the basic unit of discrete data, defined to be an item from a vocabulary indexed by {1, . . . ,V }. We represent words using unit-basis vectors that have a single component equal to one and all other components equal to zero. Thus, using superscripts to denote components, the vth word in the vocabulary is represented by a V -vector w such that wv = 1 and wu = 0 for u = v. ��� A document is a sequence of N words denoted by w = (w1, w2, . . . , wN ), where wn is the nth word in the sequence. ��� A corpus is a collection of M documents denoted by D = {w1, w2, . . . , wM }. 995

BLEI, NG, AND JORDAN We wish to find a probabilistic model of a corpus that not only assigns high probability to members of the corpus, but also assigns high probability to other ���similar��� documents. 3. Latent Dirichlet allocation Latent Dirichlet allocation (LDA) is a generative probabilistic model of a corpus. The basic idea is that documents are represented as random mixtures over latent topics, where each topic is charac- terized by a distribution over words.1 LDA assumes the following generative process for each document w in a corpus D: 1. Choose N ��� Poisson(��). 2. Choose �� ��� Dir(��). 3. For each of the N words wn: (a) Choose a topic zn ��� Multinomial(��). (b) Choose a word wn from p(wn | zn, ��), a multinomial probability conditioned on the topic zn. Several simplifying assumptions are made in this basic model, some of which we remove in subse- quent sections. First, the dimensionality k of the Dirichlet distribution (and thus the dimensionality of the topic variable z) is assumed known and fixed. Second, the word probabilities are parameter- ized by a k ��V matrix �� where ��i j = p(w j = 1 | zi = 1), which for now we treat as a fixed quantity that is to be estimated. Finally, the Poisson assumption is not critical to anything that follows and more realistic document length distributions can be used as needed. Furthermore, note that N is independent of all the other data generating variables (�� and z). It is thus an ancillary variable and we will generally ignore its randomness in the subsequent development. A k-dimensional Dirichlet random variable �� can take values in the (k - 1)-simplex (a k-vector �� lies in the (k - 1)-simplex if ��i ��� 0, ���i=1 k ��i = 1), and has the following probability density on this simplex: p(�� | ��) = �� ( ���i=1 k ��i ) ���i=1 k ��(��i) ����1-1 1 ������ ��k ��k-1, (1) where the parameter �� is a k-vector with components ��i 0, and where ��(x) is the Gamma function. The Dirichlet is a convenient distribution on the simplex ��� it is in the exponential family, has finite dimensional sufficient statistics, and is conjugate to the multinomial distribution. In Section 5, these properties will facilitate the development of inference and parameter estimation algorithms for LDA. Given the parameters �� and ��, the joint distribution of a topic mixture ��, a set of N topics z, and a set of N words w is given by: p(��, z, w | ��, ��) = p(�� | ��) N ��� n=1 p(zn | ��)p(wn | zn, ��), (2) 1. We refer to the latent multinomial variables in the LDA model as topics, so as to exploit text-oriented intuitions, but we make no epistemological claims regarding these latent variables beyond their utility in representing probability distributions on sets of words. 996

LATENT DIRICHLET ALLOCATION �� z w �� �� M N Figure 1: Graphical model representation of LDA. The boxes are ���plates��� representing replicates. The outer plate represents documents, while the inner plate represents the repeated choice of topics and words within a document. where p(zn | ��) is simply ��i for the unique i such that zn i = 1. Integrating over �� and summing over z, we obtain the marginal distribution of a document: p(w | ��, ��) = Z p(�� | ��) N ��� n=1 ��� zn p(zn | ��)p(wn | zn, ��) ! d��. (3) Finally, taking the product of the marginal probabilities of single documents, we obtain the proba- bility of a corpus: p(D | ��, ��) = M ��� d=1 Z p(��d | ��) Nd ��� n=1 ��� zdn p(zdn | ��d )p(wdn | zdn, ��) ! d��d . The LDA model is represented as a probabilistic graphical model in Figure 1. As the figure makes clear, there are three levels to the LDA representation. The parameters �� and �� are corpus- level parameters, assumed to be sampled once in the process of generating a corpus. The variables ��d are document-level variables, sampled once per document. Finally, the variables zdn and wdn are word-level variables and are sampled once for each word in each document. It is important to distinguish LDA from a simple Dirichlet-multinomial clustering model. A classical clustering model would involve a two-level model in which a Dirichlet is sampled once for a corpus, a multinomial clustering variable is selected once for each document in the corpus, and a set of words are selected for the document conditional on the cluster variable. As with many clustering models, such a model restricts a document to being associated with a single topic. LDA, on the other hand, involves three levels, and notably the topic node is sampled repeatedly within the document. Under this model, documents can be associated with multiple topics. Structures similar to that shown in Figure 1 are often studied in Bayesian statistical modeling, where they are referred to as hierarchical models (Gelman et al., 1995), or more precisely as con- ditionally independent hierarchical models (Kass and Steffey, 1989). Such models are also often referred to as parametric empirical Bayes models, a term that refers not only to a particular model structure, but also to the methods used for estimating parameters in the model (Morris, 1983). In- deed, as we discuss in Section 5, we adopt the empirical Bayes approach to estimating parameters such as �� and �� in simple implementations of LDA, but we also consider fuller Bayesian approaches as well. 997